201303.pdf 36567KB Mar 22 2013 09:11:22 PM

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MECHATRONICS IN DESIGN FRESH IDEAS ON INTEGRATING MECHANICAL SYSTEMS, ELECTRONICS, CONTROL SYSTEMS, AND SOFTWARE IN DESIGN M A S A L R I T E D E F E N S E S P R O C E S S I Y S N G Software T E M S A U T O M O Digital Control Systems Mechanical CAD C O N S T I Control Systems Mechanical Systems U M V E E R P A E R O S mechatronics Control Electronics Electromechanics R O D U P A S C T C E M E D I Electronic Systems M A F A N U C A L C T U R X E R O G I N G R A P H Y Angular velocity is often misstated Kinematic analysis requires defined reference frames and precise notation. By Kevin C Craig, PhD Ask any engineer what the velocity of a point on a rigid body is, and he will correctly say that it is the time rate of change of a linear position vector. Ask any engineer what angular velocity is, and he most likely will incorrectly say that it is the time rate of change of an angular position vector. In both cases, there is usually no mention of the reference frames involved. The misunderstanding of angular velocity—and the ambiguity that imprecise notation creates—can lead to errors that cause lost time and money, and even tragic results. A reference frame is a perspective from which observations are made regarding the motion of a system. A moving body, such as an automobile or airplane, frequently provides a useful reference frame for our observations of motion. Even when we are not moving, it is often easier to describe the motion of a point by reference to a moving object. This is the case for many common machines, such as robots. An engineer needs to be able to correlate observations of position, velocity, and acceleration of points on moving bodies, as well as the angular velocities and angular accelerations of those moving bodies, from both fixed and moving reference frames. Figure 1 R is the ground reference frame, with coordinate axes x, y, and z fixed in R. Figure 2 In a two-degree-offreedom mechanism, you can determine the angular velocity of R 2 in R, which is not a simple angular velocity. In Figure 1, R is the ground reference frame, with coordinate axes x,y, and z fixed in R. The R 1 reference frame is a rigid body moving in reference frame R, with coordinate axes x 1 , y 1 , and z 1 fixed in R 1 . β → is any vector fixed in R 1 . The defining equation for angular velocity, ω, is shown. Angular velocity is the time rate of change of orientation of the body; it is not in general equal to the derivative of any single vector. However, when a rigid body, R 1 , moves in a reference frame in such a way that there exists throughout some time interval a unit vector whose orientation in both R 1 and R is independent of the time, then R 1 is said to have simple angular velocity in Figure 3 In this serial robot, a series of links is connected by motor-actuated joints that extend from a base to an end effector. R throughout this time interval, and the magnitude of this angular-velocity vector is the time derivative of an angle. Figure 2 shows a two-degree-of-freedom mechanism. R 1 has simple angular velocity in R (ω 1 ), and R 2 has simple angular velocity in R 1 (ω 2 ). The addition theorem for angular velocity, R ω →R N= R ω →R 1+ R 1 ω →R 2+ ... + R N-1 ω →R N, allows you to easily determine the angular velocity of R 2 in R, which is not a simple angular velocity. This theorem is powerful, as it allows you to develop an expression for a complicated angular velocity by using intermediate reference frames—real or fictitious—that have simple-angular-velocity relationships between each of them. Note that there is no similar addition theorem for angular accelerations. Figure 3 shows a serial robot; that is, a series of links connected by motor-actuated joints that extend from a base to an end effector. Each link has a simple-angular-velocity relationship with the preceding link. These relationships make it easy to determine the angular velocity of a link with respect to any other link.EDN 28 EDN | MARCH <strong>2013</strong> [ www.edn.com ]
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